Poincaré Duality Embeddings and Fibrewise

We prove a relative Poincaré embedding theorem for maps of pairs into a Poincaré pair. This result has applications: it is the engine of the compression theorem of the author’s paper in Algebr. Geom. Topol. 2, (2002) 311–336, it yields a Poincaré space version of Hudson’s embedding theorem, and it can be used to equip 2-connected Poincaré spaces with handle decompositions.